LEIBNIZ FORMULAS FOR CYCLIC

HOMOTOPY FIXED POINT SPECTRA

Robert R. Bruner and John Rognes

June 2nd 2004

Abstract. We analyze the homotopy fixed point spectrum of a 𝕋-equivariant commutative S-algebra R in homological terms. There is a homological homotopy fixed point spectral sequence with E_{s,t}^{2} = H^{−s}(𝕋; H_{t}(R; 𝔽_{p})), which converges conditionally to the continuous homology H_{∗}^{c}(R^{h𝕋}; 𝔽_{p}) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations β^{ϵ}Q^{i} acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E^{2r}-term of the spectral sequence there are 2r other classes in the E^{2r}-term (obtained mostly by Dyer–Lashof operations on x) that are infinite cycles, i.e., survive to the E^{∞}-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = T HH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups C ⊂ 𝕋, and for the Tate- and homotopy orbit spectra. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras.

1. Introduction

By an S-algebra we shall either mean one in the sense of [EKMM97], or a symmetric ring spectrum in the sense of [HSS00]. For a connective S-algebra B, such as the sphere spectrum S, the complex bordism spectrum MU or the Eilenberg– Mac Lane spectrum of the integers ℤ, the algebraic K-theory spectrum K(B) can be very well approximated by the topological cyclic homology spectrum T C(B) of [BHM93], by the main theorem of [Du97]. The latter spectrum is obtained from the 𝕋-equivariant topological Hochschild homology spectrum X = T HH(B) as a homotopy limit of the fixed point spectra X^{C}, indexed over finite cyclic subgroups C of the circle group 𝕋. These fixed point spectra are in turn approximated by the homotopy fixed point spectra X^{hC} = F (EC_{+}, X)^{C}, whose homotopy groups can in principle be computed by the homotopical homotopy fixed point spectral sequence

(1.1)  E_{s,t}^{2} = H^{−s}(C; π_{t}(X)) ⟹ π_{s+t}(X^{hC}) .

1991 Mathematics Subject Classification. 19D55, 55P43, 55P91, 55S12, 55T05. Key words and phrases. Homotopy fixed points, Tate spectrum, homotopy orbits, commutative

S-algebra, Dyer–Lashof operations, differentials, topological Hochschild homology, topological cyclic homology, algebraic K-theory.

Typeset by AMS-TEX

This is derived from the tower of fibrations (with limit X^{hC}) that arises from the equivariant skeleton filtration on the free contractible C-space EC, by applying homotopy groups.

Such computations presume a rather detailed knowledge of the homotopy groups π_{∗}(X) of the 𝕋-equivariant spectrum in question. For example, [HM03] and [AuR02] deal with the cases when B is the valuation ring in a local number field and the Adams summand in p-complete connective topological K-theory, respectively. In most other cases the spectral sequence (1.1) cannot be calculated, because the homotopy groups π_{∗}(X) are not sufficiently well known.

It happens much more frequently that we are familiar with the homology groups H_{∗}(X; 𝔽_{p}). Applying mod p homology, rather than homotopy, to the tower of fibrations that approximates X^{hC} leads to a homological homotopy fixed point spectral sequence

(1.2)  E_{s,t}^{2} = H^{−s}(C; H_{t}(X; 𝔽_{p})) ⟹ H_{s+t}^{c}(X^{hC}; 𝔽_{p}) .

This spectral sequence converges conditionally [Bo99] to the (inverse) limit of the resulting tower in homology, which is not H_{∗}(X^{hC}; 𝔽_{p}), but a “continuous” version H_{∗}^{c}(X^{hC}; 𝔽_{p}) of it, for homology does not usually commute with limits.

This continuous homology, considered as a comodule over the dual Steenrod algebra A_{∗} [Mi58], is nonetheless a powerful invariant of X^{hC}. In particular, when X is bounded below and of finite type there is a strongly convergent spectral sequence

(1.3)  E_{2}^{s,t} = Ext_{A_{∗}}^{s,t}(𝔽_{p}, H_{∗}^{c}(X^{hC}; 𝔽_{p})) ⟹ π_{t−s}(X^{hC})_{p}^{∧}

which can be obtained as an inverse limit of Adams spectral sequences [CMP87, 7.1]. Hence the continuous homology does in some sense determine the p-adic homotopy type of X^{hC}.

A form of the spectral sequence (1.3) was most notably applied in the proofs by W.H. Lin [LDMA80] and J. Gunawardena [AGM85] of the Segal conjecture for cyclic groups of prime order. The conjecture corresponds to the special case of the discussion above when B = S is the sphere spectrum, so X = T HH(S) = S is the 𝕋-equivariant sphere spectrum, which is split [LMS86, II.8] so that X^{hC} ≃ F (BC_{+}, S) = D(BC_{+}). The proven Segal conjecture [Ca84] then tells us that for each p-group C the comparison map X^{C} → X^{hC} is a p-adic equivalence. The proof of the general (cyclic) case is by reduction to the initial case when C = C_{p} is of prime order, and therefore relies on the theorems of Lin and Gunawardena cited above. In this case, of course, we do not know π_{∗}(X) = π_{∗}(S) sufficiently well, but H_{∗}(X; 𝔽_{p}) = 𝔽_{p} is particularly simple. The proof of the theorems of Lin and Gunawardena now amounts to showing that although the natural homomorphism H_{∗}(X^{C}; 𝔽_{p}) → H_{∗}^{c}(X^{hC}; 𝔽_{p}) of A_{∗}-comodules is not in itself an isomorphism, it does induce an isomorphism of E_{2}-terms upon applying the functor Ext_{A_{∗}}^{∗∗}(𝔽_{p}, −).

Returning to the general situation, we are therefore interested in studying (i) the differentials in the homological homotopy fixed point spectral sequence (1.2) above, and (ii) the A_{∗}-comodule extension questions relating its E^{∞}-term to the abutment H_{∗}^{c}(X^{hC}; 𝔽_{p}). There will in general be non-trivial differentials in (1.2), but our main Theorem 1.5 below provides a very general and useful collapse result, as is illustrated by the examples in Section 6. The identification of the A_{∗}-comodule structure on the abutment plays a crucial role already in the case X = S, but requires further study beyond that given here, and will be presented in the forthcoming Ph.D. thesis of Sverre Lunøe–Nielsen [L-N].

When B is a commutative S-algebra then so is X = T HH(B), and the tower of fibrations with limit X^{hC} is one of commutative S-algebras [EKMM97, IX]. Therefore there are Dyer–Lashof operations acting on the spectral sequence (1.2) in this case, rather analogously to the action by Steenrod operations in the Adams spectral sequence of a commutative S-algebra [BMMS86, IV]. In the latter case there are interesting relations between the Adams differentials and the Steenrod operations, which propagate early differentials to higher degrees. The initial motivation for the present article was to determine the analogous interaction between the differentials and the Dyer–Lashof operations in the homological homotopy fixed point spectral sequence of a commutative S-algebra, hereafter denoted X = R. However, the analogy with the behavior of differentials in the Adams spectral sequence is more apparent than real, suggesting neither the survival to E^{∞} of some classes nor the method of proof. In particular, there is no analog in the Adams spectral sequence of our main collapse result, Theorem 1.5.

For each finite subgroup C ⊂ 𝕋 the homological spectral sequence for R^{hC} is an algebra over the corresponding homological spectral sequence for R^{h𝕋}, as outlined in Section 7, so it will suffice for us to consider the circle homotopy fixed points R^{h𝕋} and the case C = 𝕋 of the spectral sequence (1.2). Our first results in Sections 2–4 can then be summarized as follows.

Theorem 1.4. (a) Let R be a 𝕋-equivariant commutative S-algebra. There is a natural A_{∗}-comodule algebra spectral sequence

E_{∗∗}^{2} = H^{−∗}(𝕋; H_{∗}(R; 𝔽_{p})) = P (y) ⊗ H_{∗}(R; 𝔽_{p})

with y in bidegree (−2, 0), which is conditionally convergent to the continuous homology

H_{∗}^{c}(R^{h𝕋}; 𝔽_{p}) = lim_{n} H_{∗}(F (S_{+}^{2n+1}, R)^{𝕋}; 𝔽_{p})

of the homotopy fixed point spectrum R^{h𝕋}.

(b) There are natural Dyer–Lashof operations β^{ϵ}Q^{i} acting vertically on this homological homotopy fixed point spectral sequence. For each element x ∈ E_{0,t}^{2r} ⊂ H_{t}(R; 𝔽_{p}) we have the relation

d^{2r}(β^{ϵ}Q^{i}(x)) = β^{ϵ}Q^{i}(d^{2r}(x))

for every integer i and ϵ ∈ {0, 1}. If d^{2r}(x) = y^{r} · δx with δx ∈ H_{t+2r−1}(R; 𝔽_{p}), the right hand side β^{ϵ}Q^{i}(d^{2r}(x)) is y^{r} · β^{ϵ}Q^{i}(δx).

(c) The classes y^{n} are infinite cycles, so the differentials from the vertical axis E_{0,∗}^{2r} propagate to each column by the relation

d^{2r}(y^{n} · x) = y^{n} · d^{2r}(x)

for all x ∈ E_{0,∗}^{2r}, 2r ≥ 2, n ≥ 0. Hence there are isomorphisms E_{∗∗}^{2r} ≡ P (y) ⊗ E_{0,∗}^{2r} for all 2r ≥ 2, modulo y-torsion in filtrations −2r < s ≤ 0.

For proofs, see Proposition 2.4, Proposition 4.1 and Lemma 4.3. The key idea is to identify the differentials in the homological homotopy fixed point spectral sequence as obstructions to extending equivariant maps, as explained in Section 3. Note that the spectral sequence is concentrated in even columns, so all differentials of odd length (d^{r} with r odd) must vanish.

Our main theorem is the following collapse result.

Theorem 1.5. Let R be a 𝕋-equivariant commutative S-algebra, suppose that x ∈ H_{t}(R; 𝔽_{p}) survives to the E^{2r}-term E_{0,t}^{2r} ⊂ H_{t}(R; 𝔽_{p}) of the homological homotopy fixed point spectral sequence for R and write d^{2r}(x) = y^{r} · δx.

(a) For p = 2, the 2r classes

x^{2} = Q^{t}(x), Q^{t+1}(x), …, Q^{t+2r−2}(x) and Q^{t+2r−1}(x) + xδx

all survive to the E^{∞}-term, i.e., are infinite cycles.

(b) For p odd and t = 2m even, the 2r classes

x^{p} = Q^{m}(x), βQ^{m+1}(x), …, Q^{m+r−1}(x) and x^{p−1}δx

all survive to the E^{∞}-term, i.e., are infinite cycles.

(c) For p odd and t = 2m − 1 odd, the 2r classes

βQ^{m}(x), Q^{m}(x), …, βQ^{m+r−1}(x) and Q^{m+r−1}(x) − x(δx)^{p−1}

all survive to the E^{∞}-term, i.e., are infinite cycles.

This is proved in Section 5 as our Theorem 5.1. To be perfectly clear, in case (a) the classes are x^{2} = Q^{t}(x), Q^{i}(x) for t + 1 ≤ i ≤ t + 2r − 2, and Q^{t+2r−1}(x) + xδx, in case (b) the classes are x^{p} = Q^{m}(x), β^{ϵ}Q^{i}(x) for m + 1 ≤ i ≤ m + r − 1 and ϵ ∈ {0, 1}, and x^{p−1}δx, and in case (c) the classes are β^{ϵ}Q^{i}(x) for m ≤ i ≤ m+r−2 and ϵ ∈ {0, 1}, βQ^{m+r−1}(x), and Q^{m+r−1}(x) − x(δx)^{p−1}.

There are similar extensions of our results to the Tate constructions R^{tC} = [ẼC ∧ F (EC_{+}, R)]^{C} and homotopy orbit spectra R_{hC} = EC_{+} ∧_{C} R, but to keep the exposition simple these are also only discussed in Section 7.

As applications of our main results, we turn in Section 6 to the study of the algebraic K-theory spectrum K(MU) which interpolates between K(S) (which is Waldhausen’s A(∗), related to high dimensional geometric topology) and K(ℤ) (which relates to the Vandiver and Leopoldt conjectures, and other number theory), by the methods of topological cyclic homology. Hence we must study the fixed- and homotopy fixed point spectra of the commutative S-algebra R = T HH(MU), for various subgroups C of the circle group 𝕋. It is known that H_{∗}(MU; 𝔽_{p}) = P (b_{k} | k ≥ 1), where P (−) denotes the polynomial algebra over 𝔽_{p} and |b_{k}| = 2k, from which it follows ([MS93, 4.3] or [CS]) that H_{∗}(T HH(MU); 𝔽_{p}) = H_{∗}(MU; 𝔽_{p}) ⊗ E(σb_{k} | k ≥ 1), where E(−) denotes the exterior algebra over 𝔽_{p} and σ: H_{∗}(R; 𝔽_{p}) → H_{∗+1}(R; 𝔽_{p}) is the degree +1 operator induced by the circle action. Hence the homological homotopy fixed point spectral sequence for T HH(MU)^{h𝕋} begins

E_{∗∗}^{2} = P (y) ⊗ P (b_{k} | k ≥ 1) ⊗ E(σb_{k} | k ≥ 1) .

There are differentials d^{2}(b_{k}) = y · σb_{k} for all k ≥ 1, so by our Theorem 1.4

E_{∗∗}^{4} = P (y) ⊗ P (b_{k}^{p} | k ≥ 1) ⊗ E(b_{k}^{p−1}σb_{k} | k ≥ 1)

plus some classes (the image of σ) in filtration s = 0. By our Theorem 1.5, the spectral sequence collapses completely at the E^{4}-term, so that

H_{∗}^{c}(T HH(MU)^{h𝕋}; 𝔽_{p}) = P (y) ⊗ P (b_{k}^{p} | k ≥ 1) ⊗ E(b_{k}^{p−1}σb_{k} | k ≥ 1)

plus some classes in filtration zero, as an algebra. The identification of the A_{∗}- comodule extensions remains, for which we refer to the cited Ph.D. thesis [L-N]. This provides the input for the inverse limit of Adams spectral sequences (1.3) converging to π_{∗}(T HH(MU)^{h𝕋})_{p}^{∧}, which approximates the topological version T F (MU) of negative cyclic homology, and which determines the topological cyclic homology of MU by a fiber sequence

T C(MU) ⟶^{π} T F (MU) ⟶^{R−1} T F (MU) .

The fiber of the cyclotomic trace map K(MU) → T C(MU) is equivalent to that of K(ℤ) → T C(ℤ), by [Du97], which now is quite well known [Ro02], [Ro03]. Our theorem therefore provides a key input to the computation of K(MU). See Theorem 6.4(a).

Similar applications are given for the connective Johnson–Wilson spectra B = BP ⟨n⟩, for p and n such that these are commutative S-algebras, and the (higher real) commutative S-algebras B = ko and tmf for p = 2. See Section 6. Lastly, we can also show the collapse at E_{∗∗}^{4} of the homological homotopy fixed point spectral sequence for R = T HH(BP), where BP is the p-local Brown–Peterson S-algebra [BJ02], without making the (presently uncertain) assumption that BP can be realized as a commutative S-algebra. See Theorem 6.4(b). This is possible by the homological approach, since the split surjection H_{∗}(MU; 𝔽_{p}) → H_{∗}(BP; 𝔽_{p}) prevails throughout the homological spectral sequences.

2. A homological spectral sequence

Let 𝕋 ⊂ ℂ^{∗} be the circle group. As our model for a free contractible 𝕋-CW complex E𝕋 we take the unit sphere S^{∞} ⊂ ℂ^{∞} with the usual coordinatewise action by 𝕋. It has one 𝕋-equivariant cell in each even non-negative dimension. The equivariant 2n-skeleton is the odd sphere E𝕋^{(2n)} = S^{2n+1} ⊂ ℂ^{n+1}, which is obtained from the equivariant (2n − 2)-skeleton E𝕋^{(2n−2)} = S^{2n−1} ⊂ ℂ^{n} by attaching a free 𝕋-equivariant 2n-cell 𝕋 × D^{2n} along the group action map α: 𝕋 × S^{2n−1} → S^{2n−1}. Hence there is a 𝕋-equivariant filtration

(2.1)  ∅ ⊂ S^{1} ⊂ ⋯ ⊂ S^{2n−1} ⊂ S^{2n+1} ⊂ …

with colimit E𝕋, and 𝕋-equivariant cofiber sequences

S^{2n−1} → S^{2n+1} → 𝕋_{+} ∧ S^{2n}

for each n ≥ 0.

Let X be any spectrum with 𝕋-action, i.e., a naively 𝕋-equivariant spectrum. The homotopy fixed point spectrum of X is defined as the mapping spectrum

X^{h𝕋} = F (E𝕋_{+}, X)^{𝕋}

of 𝕋-equivariant based maps from E𝕋_{+} to X. The filtration (2.1) of E𝕋 = S^{∞} induces a tower of fibrations

(2.2)  ⋯ → F (S_{+}^{2n+1}, X)^{𝕋} → F (S_{+}^{2n−1}, X)^{𝕋} → ⋯ → F (S_{+}^{1}, X)^{𝕋} = X → ∗

with the homotopy fixed point spectrum as its (homotopy) limit

X^{h𝕋} ≃ holim_{n} F (S_{+}^{2n+1}, X)^{𝕋} .

The cofiber sequences above induce (co-)fiber sequences of spectra

Σ^{−2n}X = F (𝕋_{+} ∧ S^{2n}, X)^{𝕋} → F (S_{+}^{2n+1}, X)^{𝕋} → F (S_{+}^{2n−1}, X)^{𝕋}

for each n ≥ 0.

We now place F (S_{+}^{2n−1}, X)^{𝕋} in the two filtrations s = −2n and s = −2n + 1, for each n ≥ 0. Hence we obtain a chain of cofiber sequences of spectra:

\xymatrix{F (S_{+}^{2n+1}, X)^{𝕋} \ar[r] & F (S_{+}^{2n−1}, X)^{𝕋} \ar[d] \ar[r] & F (S_{+}^{2n−1}, X)^{𝕋} \ar[d] \ar[r] & F (S_{+}^{2n−3}, X)^{𝕋} \ar[d] \\ … & Σ^{−2n}X \ar[ul] & ∗ \ar[ul] & Σ^{−2n+2}X \ar[ul]}

Here the filtrations −2n − 1 ≤ s ≤ −2n + 2 are displayed.

Next we apply mod p homology to this chain of cofiber sequences, to obtain a homologically indexed unrolled exact couple [Bo99, 0.1] with

A_{s,t} = H_{s+t}(F (S_{+}^{2n−1}, X)^{𝕋}; 𝔽_{p})

for s = −2n and s = −2n + 1, and

E_{s,t} = H_{s+t}(Σ^{−2n}X; 𝔽_{p}) = H_{t}(X; 𝔽_{p})

for s = −2n and zero otherwise. Here E_{s,t} = E_{s,t}^{1} = E_{s,t}^{2}.

The E^{2}-term of the associated spectral sequence can be expressed as the group cohomology

E_{s,t}^{2} = H^{−s}(𝕋; H_{t}(X; 𝔽_{p})) ≅ H^{−s}(𝕋; 𝔽_{p}) ⊗ H_{t}(X; 𝔽_{p})

of the circle group 𝕋, acting trivially on H_{∗}(X; 𝔽_{p}) as it must since 𝕋 is path connected. We have H^{∗}(𝕋; 𝔽_{p}) = P (y) with y in degree 2, where P (−) denotes the polynomial algebra, so

E_{∗∗}^{2} = P (y) ⊗ H_{∗}(X; 𝔽_{p})

with y in bidegree (−2, 0) and H_{t}(X; 𝔽_{p}) in bidegree (0, t). See [GM95, 14.2] for a discussion of this and related spectral sequences.

Since A_{s} = 0 for s ≥ 0 we have A_{∞} = colim_{s} A_{s} = 0. Therefore the spectral sequence is conditionally convergent, by [Bo99, 5.10], in this case to the limit A_{−∞} = lim_{s} A_{s}. Indexing the limit system by n in place of s, it can be written as

(2.3)  H_{∗}^{c}(X^{h𝕋}; 𝔽_{p}) = lim_{n} H_{∗}(F (S_{+}^{2n+1}, X)^{𝕋}; 𝔽_{p}) ,

which we call the continuous homology of X^{h𝕋}. The spectral sequence will be strongly convergent to this target if the criterion RE_{∗∗}^{∞} = 0 of [Bo99, 7.4] is satisfied, for which it suffices that in each bidegree (s, t) we have E_{s,t}^{r} = E_{s,t}^{∞} for some finite r = r(s, t).

Proposition 2.4. There is a natural spectral sequence

E_{∗∗}^{2} = H^{−∗}(𝕋; H_{∗}(X; 𝔽_{p})) = P (y) ⊗ H_{∗}(X; 𝔽_{p})

with y in bidegree (−2, 0), which is conditionally convergent to the continuous homology H_{∗}^{c}(X^{h𝕋}; 𝔽_{p}). We call this the homological homotopy fixed point spectral sequence. If H_{∗}(X; 𝔽_{p}) is finite in each degree, or the spectral sequence collapses at a finite stage, then the spectral sequence is strongly convergent.

Remark 2.5. Since homology does usually not commute with the formation of limits, the canonical map

H_{∗}(X^{h𝕋}; 𝔽_{p}) → H_{∗}^{c}(X^{h𝕋}; 𝔽_{p})

is usually not an isomorphism. The Segal conjecture provides striking examples of this phenomenon.

As noted in the introduction, it is rather more traditional to apply the homotopy group functor π_{∗} to the tower of fibrations (2.2), to obtain an unraveled exact couple and a conditionally convergent (homotopical) homotopy fixed point spectral sequence

E_{s,t}^{2} = H^{−s}(𝕋; π_{t}(X)) ⟹ π_{s+t}(X^{h𝕋}) .

However, this is not the spectral sequence that we will consider. Recent work by Ch. Ausoni and the second author [AuR02, §4], as well as current work by S. Lunøe-Nielsen (and the second author) [L-N] support the assertion that the homological spectral sequence is an interesting object.

3. Differentials

We now make the differentials in the homological homotopy fixed point spectral sequence more explicit, as obstructions to extending equivariant maps.

Consider a class x ∈ H_{t}(X; 𝔽_{p}), represented at the E^{2}-term of the homological spectral sequence in bidegree (0, t). Let H = H𝔽_{p} be the mod p Eilenberg–Mac Lane spectrum. Then x can be represented as a non-equivariant map S^{t} → H ∧ X, or equivalently as a 𝕋-equivariant map

x: S_{+}^{1} ∧ S^{t} → H ∧ X .

Here 𝕋 acts on S_{+}^{1} (freely off the base point) and X, but not on S^{t} or H.

The condition that x ∈ E_{0,t}^{2} = H_{t}(X; 𝔽_{p}) survives to the E^{2r}-term, i.e., that all differentials d^{2}(x), …, d^{2r−2}(x) vanish, is equivalent to x being in the image from H_{t}(F (S_{+}^{2r−1}, X)^{𝕋}; 𝔽_{p}) under the map induced by restriction along S_{+}^{1} ⊂ S_{+}^{2r−1}. This is in turn equivalent to the existence of a 𝕋-equivariant extension

x′: S_{+}^{2r−1} ∧ S^{t} → H ∧ X

of x along S_{+}^{1} ⊂ S_{+}^{2r−1}, in view of the natural equivariant equivalence

H ∧ F (S_{+}^{2r−1}, X) ⟶^{≃} F (S_{+}^{2r−1}, H ∧ X) .

(To establish this equivalence, note that the finite 𝕋-CW complex S_{+}^{2r−1} admits a 𝕋-equivariant Spanier–Whitehead dual. We are considering maps from free 𝕋-CW complexes into these spectra, so only the naive notion of a 𝕋-equivariant equivalence is required.)

Suppose that x ∈ E_{0,t}^{2r} has survived to the E^{2r}-term, so that such a 𝕋-equivariant extension x′ exists. Then the differential

d^{2r}(x) ∈ E_{−2r,t+2r−1}^{2r}

is the obstruction to extending x′ further along S_{+}^{2r−1} ⊂ S_{+}^{2r+1} to an equivariant map

x′′: S_{+}^{2r+1} ∧ S^{t} → H ∧ X .

We put the obvious right adjoints of these maps together in a diagram, as below.

\xymatrix{ & S_{+}^{1} \ar[dr]^{x} \ar[d] \\ (𝕋 × S^{2r−1})_{+} \ar[r]^{α_{+}} \ar[d] & S_{+}^{2r−1} \ar[r]^{x′} \ar[d] & F (S^{t}, H ∧ X) \\ (𝕋 × D^{2r})_{+} \ar[r] & S_{+}^{2r+1} \ar@{-->}[ur]_{x′′}}

But since S_{+}^{2r+1} is obtained from S_{+}^{2r−1} by adjoining a free 𝕋-cell along the action map α: 𝕋 × S^{2r−1} → S^{2r−1}, the obstruction to such an extension is precisely the obstruction to extending the equivariant map x′ ∘ α_{+} from (𝕋 × S^{2r−1})_{+} over (𝕋×D^{2r})_{+}. Equivalently, the obstruction is that of finding a homotopy to a constant map of the non-equivariant map x̄: S^{2r−1} → F (S^{t}, H ∧ X) given by regarding x′ as a non-equivariant map, and then restricting away from the disjoint base point. Its left adjoint again is then a map

x̄: S^{2r−1} ∧ S^{t} → H ∧ X .

We summarize:

Lemma 3.1. Let x ∈ E_{0,t}^{2r} ⊂ H_{t}(X; 𝔽_{p}) be represented by a 𝕋-equivariant map x: S_{+}^{1} ∧S^{t} → H ∧X, which extends to an equivariant map x′: S_{+}^{2r−1} ∧S^{t} → H ∧X. Then d^{2r}(x) = y^{r} ·x̄, where x̄ ∈ H_{t+2r−1}(X; 𝔽_{p}) is represented by x′ considered as a non-equivariant map, restricted to the stable summand S^{2r−1}∧S^{t} of S_{+}^{2r−1}∧S^{t}. □

The extended map x′ represents a class in the homology of F (S_{+}^{2r−1}, X)^{𝕋}, and considering x′ as a non-equivariant map amounts to following the map

φ: F (S_{+}^{2r−1}, X)^{𝕋} → F (S_{+}^{2r−1}, X)

that forgets the 𝕋-equivariance. There is a canonical map

ν: DS_{+}^{2r−1} ∧ X → F (S_{+}^{2r−1}, X)

where DS_{+}^{2r−1} = F (S_{+}^{2r−1}, S) is the functional dual of S_{+}^{2r−1}, and ν is a weak equivalence by Spanier–Whitehead duality, since S_{+}^{2r−1} is a finite CW complex. See [LMS86, §III.1]. Hence there is a natural isomorphism

ν: H^{−∗}(S^{2r−1}; 𝔽_{p}) ⊗ H_{∗}(X; 𝔽_{p}) → H_{∗}(F (S_{+}^{2r−1}, X); 𝔽_{p})

where we have identified H_{∗}(DS_{+}^{2r−1}; 𝔽_{p}) with H̃^{−∗}(S_{+}^{2r−1}; 𝔽_{p}) = H^{−∗}(S^{2r−1}; 𝔽_{p}). We write H^{∗}(S^{2r−1}; 𝔽_{p}) = E(ι_{2r−1}), where ι_{2r−1} is the canonical generator in degree (2r − 1) and E(−) denotes the exterior algebra.

Proposition 3.2. The composite map

H_{∗}(F (S_{+}^{2r−1}, X)^{𝕋}; 𝔽_{p}) ⟶^{φ_{∗}} H_{∗}(F (S_{+}^{2r−1}, X); 𝔽_{p})

⟵_{≅}^{ν_{∗}} H^{−∗}(S^{2r−1}; 𝔽_{p}) ⊗ H_{∗}(X; 𝔽_{p})

takes any class x′ that is mapped to x ∈ E_{0,t}^{2r} ⊂ H_{t}(X; 𝔽_{p}) by the restriction map

H_{∗}(F (S_{+}^{2r−1}, X)^{𝕋}; 𝔽_{p}) → H_{∗}(F (S_{+}^{1}, X)^{𝕋}; 𝔽_{p}) = H_{∗}(X; 𝔽_{p})

to the sum

(ν_{∗}^{−1}φ_{∗})(x′) = 1 ⊗ x + ι_{2r−1} ⊗ δx ,

where d^{2r}(x) = y^{r} ·δx in E_{−2r,t+2r−1}^{2r}. Suppressing the power of y we may somewhat imprecisely write this formula as

φ_{∗}(x) = 1 ⊗ x + ι_{2r−1} ⊗ d^{2r}(x) .

The case r = 1 says d^{2}(x) = y · σx, and follows e.g. from [Ro98, 3.3].

Proof. This is really a corollary to Lemma 3.1, but for the observation that the restriction of the non-equivariant x′ to the subspace S^{t} ⊂ S_{+}^{2r−1} ∧ S^{t} equals the restriction of the non-equivariant x to the same subspace S^{t} ⊂ S_{+}^{1} ∧ S^{t}, which in turn corresponds to x ∈ E_{0,t}^{2r} under the identification H_{∗}(F (S_{+}^{1}, X)^{𝕋}; 𝔽_{p}) = H_{∗}(X; 𝔽_{p}). □

Remark 3.3. Lemma 3.1 says that the differential in the homotopy fixed point spectral sequence is essentially the 𝕋-equivariant root invariant for H ∧ X. A corresponding description of the (Mahowald) C_{2}-equivariant root invariant for S can be found in [BG95, 2.5]: Let S^{n+kα} denote the C_{2}-equivariant sphere that is the one point compactification of ℝ^{n} ⊕ ℝ^{k}(−1), where C_{2} acts trivially on ℝ^{n} and by negation on ℝ^{k}(−1). Given a non-equivariant (stable) map x : S^{n} → S^{0}, let x′ : S^{n+kα} → S^{0} be a C_{2}-equivariant extension of x with k maximal. Then the C_{2}-equivariant root invariant of x contains the non-equivariant map x′ : S^{n+k} → S^{0} underlying x′.

4. Commutative S-algebras

Now suppose that X = R is a (naively) 𝕋-equivariant commutative S-algebra, i.e., a commutative S-algebra with a continuous point-set level action by the circle group 𝕋. We shall be concerned with the homotopy fixed points of R, rather than its genuine fixed points, so only this weak notion of an equivariant spectrum will be needed [GM95, §1]. Our principal example is R = T HH(B), the topological Hochschild homology spectrum of another commutative S-algebra B. The cyclic structure on topological Hochschild homology then provides the relevant 𝕋-action [EKMM97, IX].

In this situation the homotopy fixed point spectrum R^{h𝕋} = F (E𝕋_{+}, R)^{𝕋} is also a commutative S-algebra. Writing μ: R∧R → R for the 𝕋-equivariant multiplication map of R, the corresponding multiplication map for R^{h𝕋} is given by the composite

F (E𝕋_{+}, R)^{𝕋} ∧ F (E𝕋_{+}, R)^{𝕋} ⟶^{∧} F (E𝕋_{+} ∧ E𝕋_{+}, R ∧ R)^{𝕋}

⟶^{μ_{#}Δ^{#}} F (E𝕋_{+}, R)^{𝕋} .

Here ∧ smashes together two 𝕋-equivariant maps Σ^{∞}E𝕋_{+} → R, and considers the resulting (𝕋 × 𝕋)-equivariant map as a 𝕋-equivariant map by the diagonal action. The map μ_{#} composes on the left by μ: R ∧ R → R, while the map Δ^{#} composes on the right by the space level diagonal map Δ: E𝕋_{+} → E𝕋_{+} ∧ E𝕋_{+}. Since μ is commutative and Δ is cocommutative, the resulting multiplication on R^{h𝕋} is also strictly commutative.

Writing η: S → R for the 𝕋-equivariant unit map of R, the corresponding unit map for R^{h𝕋} is the composite

S → F (E𝕋_{+}, S)^{𝕋} ⟶^{η_{#}} F (E𝕋_{+}, R)^{𝕋} .

Here the definition of the first map relies on the fact that 𝕋 acts trivially on S.

Commutative S-algebras are E_{∞} ring spectra, and are in particular also H_{∞} ring spectra. Hence there are Dyer–Lashof operations Q^{i} acting on their mod p homology algebras [BMMS86, §III.1]. Recall that Q^{i} is a natural transformation

Q^{i}: H_{t}(R; 𝔽_{p}) → H_{t+iq}(R; 𝔽_{p})

for all integers t, where q = 2p − 2. We also include their composites βQ^{i} with the homology Bockstein operation β: H_{t}(R; 𝔽_{p}) → H_{t−1}(R; 𝔽_{p}) as generators of the Dyer–Lashof algebra. For p = 2 the standard notation is to write Q^{2i} and Q^{2i−1} for the operations that would otherwise be called Q^{i} and βQ^{i}, respectively.

The homological homotopy fixed point spectral sequence of Proposition 2.4 is derived by applying homology to the tower (2.2). Now that X = R, each spectrum F (S_{+}^{2n+1}, R)^{𝕋} is a commutative S-algebra, for the same reasons as we just indicated for R^{h𝕋}, and each fibration in the tower is a map of commutative S-algebras. Therefore the spectral sequence is one of commutative (A_{∗}-comodule) algebras over the Dyer–Lashof algebra. We can make this action quite explicit, as follows.

Proposition 4.1. Let R be a 𝕋-equivariant commutative S-algebra, and let E_{∗∗}^{r} be its homological homotopy fixed point spectral sequence. Then for each element x ∈ E_{0,t}^{2r} ⊂ H_{t}(R; 𝔽_{p}) we have the relation

d^{2r}(β^{ϵ}Q^{i}(x)) = β^{ϵ}Q^{i}(d^{2r}(x)) ,

for every integer i and ϵ ∈ {0, 1}. Here the right hand side should be interpreted as follows. If d^{2r}(x) = y^{r} · δx with δx ∈ H_{t+2r−1}(R; 𝔽_{p}) then β^{ϵ}Q^{i}(d^{2r}(x)) = y^{r} · β^{ϵ}Q^{i}(δx).

The case r = 1 also appears as [AnR, 5.9].

Proof. Let x ∈ H_{t}(R; 𝔽_{p}) and suppose that x survives to the E^{2r}-term. Then there exists an extension x′ ∈ H_{t}(F (S_{+}^{2r−1}, R)^{𝕋}; 𝔽_{p}) of x over the restriction map, and z′ = β^{ϵ}Q^{i}(x′) is an extension of z = β^{ϵ}Q^{i}(x) over the same map, by naturality. The maps φ and ν from Proposition 3.2 are both maps of commutative S-algebras, and therefore induce (A_{∗}-comodule) algebra homomorphisms φ_{∗} and ν_{∗} that commute with the Dyer–Lashof operations. Thus

(4.2)  (ν_{∗}^{−1}φ_{∗})(β^{ϵ}Q^{i}(x′)) = 1 ⊗ β^{ϵ}Q^{i}(x) + ι_{2r−1} ⊗ δz

where d^{2r}(β^{ϵ}Q^{i}(x)) = y^{r} · δz, is equal to

β^{ϵ}Q^{i}((ν_{∗}^{−1}φ_{∗})(x′)) = β^{ϵ}Q^{i}(1 ⊗ x + ι_{2r−1} ⊗ δx)

where d^{2r}(x) = y^{r} · δx. Now the Dyer–Lashof operations on the homology of the smash product DS_{+}^{2r−1} ∧R are given by a Cartan formula, and on the tensor factor H_{∗}(DS_{+}^{2r−1}; 𝔽_{p}) ≅ H^{−∗}(S^{2r−1}; 𝔽_{p}) the operation β^{ϵ}Q^{i} corresponds to the Steenrod operation β^{ϵ}P^{−i}, by [BMMS86, III.1.2]. But the latter operations all act trivially on H^{∗}(S^{2r−1}; 𝔽_{p}), except for P^{0} = 1, so the Cartan formula gives

β^{ϵ}Q^{i}(1 ⊗ x + ι_{2r−1} ⊗ δx) = 1 ⊗ β^{ϵ}Q^{i}(x) + ι_{2r−1} ⊗ β^{ϵ}Q^{i}(δx) .

Identifying this with (4.2) and comparing the coefficients of ι_{2r−1} we obtain the identity

δz = β^{ϵ}Q^{i}(δx) ,

as claimed. □

Any spectrum X can be considered as a module over the sphere spectrum S, and any action by 𝕋 on X may be taken to be in the category of S-modules. It follows that the homological homotopy fixed point spectral sequence for X is a module over the corresponding spectral sequence for S, which is an algebra spectral sequence by our previous remarks (since S is a commutative S-algebra).

In fact the homological homotopy fixed point spectral sequence for S is particularly simple, since H_{∗}(S; 𝔽_{p}) = 𝔽_{p} is concentrated in degree 0, so the spectral sequence collapses to

E_{∗∗}^{2} = P (y) ,

which is concentrated on the horizontal axis. Hence each power of y is an infinite cycle, i.e., d^{r}(y^{n}) = 0 for all r and n.

Since the spectral sequence for X is a module over the one for S, the Leibniz formula for the module pairing immediately yields the following result.

Lemma 4.3. Let X be any 𝕋-equivariant spectrum. The differentials in the homological homotopy fixed point spectral sequence converging to H_{∗}^{c}(X^{h𝕋}; 𝔽_{p}) satisfy the relation

d^{2r}(y^{n} · x) = y^{n} · d^{2r}(x)

for all x ∈ E_{0,∗}^{2r} ⊂ H_{∗}(X; 𝔽_{p}), 2r ≥ 2 and n ≥ 0. Hence the spectral sequence is completely determined by the differentials that originate on the vertical axis. □

Remark 4.4. A proof by induction on r shows that each class in E_{−2n,t}^{2r} has the form y^{n} · x for a class x ∈ E_{0,t}^{2r} ⊂ H_{t}(X; 𝔽_{p}). The E^{2r}-term may therefore only contain y-torsion of height strictly less than r, and concentrated in filtrations −2r < s ≤ 0. In Section 7 we shall remark on an analogous homological Tate spectral sequence, where P (y) is replaced by P (y, y^{−1}) and the issue of y-torsion classes becomes void.

5. Infinite cycles

The Dyer–Lashof operations satisfy instability conditions [BMMS86, III.1.1] that are in a sense dual to those of the Steenrod operations. For a class x ∈ H_{t}(R; 𝔽_{p}) the lowest nontrivial operation is Q^{t}(x) = x^{2} when p = 2, Q^{m}(x) = x^{p} when p is odd and t = 2m is even, and βQ^{m}(x) when p is odd and t = 2m − 1 is odd. Similarly, the lowest nontrivial operation on δx ∈ H_{t+2r−1}(R; 𝔽_{p}) with d^{2r}(x) = y^{r} · δx is Q^{t+2r−1}(δx) = (δx)^{2} when p = 2, βQ^{m+r}(δx) when p is odd and t = 2m is even, and Q^{m+r−1}(δx) = (δx)^{p} when p is odd and t = 2m − 1 is odd. Thus there is in each case a sequence of (2r − 1) Dyer–Lashof operations β^{ϵ}Q^{i} whose action on x can be nontrivial, but whose action on δx must be trivial. By Proposition 4.1, this sequence of operations on x will survive past the E^{2r}-term, at least to the E^{2r+2}-term. It is the main point of the present article to show that these classes, and one more “companion class”, then in fact go on indefinitely to survive to the E^{∞}-term, i.e., are infinite cycles!

Theorem 5.1. Let R be a 𝕋-equivariant commutative S-algebra, suppose that x ∈ H_{t}(R; 𝔽_{p}) survives to the E^{2r}-term E_{0,t}^{2r} ⊂ H_{t}(R; 𝔽_{p}) of the homological homotopy fixed point spectral sequence for R, and write d^{2r}(x) = y^{r} · δx.

(a) For p = 2, the 2r classes

x^{2} = Q^{t}(x), Q^{t+1}(x), …, Q^{t+2r−2}(x) and Q^{t+2r−1}(x) + xδx

all survive to the E^{∞}-term, i.e., are infinite cycles.

(b) For p odd and t = 2m even, the 2r classes

x^{p} = Q^{m}(x), βQ^{m+1}(x), …, Q^{m+r−1}(x) and x^{p−1}δx

all survive to the E^{∞}-term, i.e., are infinite cycles.

(c) For p odd and t = 2m − 1 odd, the 2r classes

βQ^{m}(x), Q^{m}(x), …, βQ^{m+r−1}(x) and Q^{m+r−1}(x) − x(δx)^{p−1}

all survive to the E^{∞}-term, i.e., are infinite cycles.

Proof. The argument proceeds by considering a universal example. Recall that a class x ∈ E_{0,t}^{2r} is represented by a 𝕋-equivariant map x: S_{+}^{1} ∧ S^{t} → H ∧ R that admits an equivariant extension x′: S_{+}^{2r−1} ∧ S^{t} → H ∧ R. Let

X = D_{p}(S_{+}^{2r−1} ∧ S^{t}) = EΣ_{p} ⋉_{Σ_{p}} (S_{+}^{2r−1} ∧ S^{t})^{∧p}

be the p-th extended power of S_{+}^{2r−1} ∧ S^{t}.

Somewhat abusively, we write H̃_{∗}(S_{+}^{2r−1} ∧ S^{t}; 𝔽_{p}) = 𝔽_{p}{x, δx} with |x| = t and |δx| = t + 2r − 1. Then the homology of the p-th extended power is

H_{∗}(X; 𝔽_{2}) = 𝔽_{2}{xδx, Q^{i}(x) | i ≥ t, Q^{i}(δx) | i ≥ t + 2r − 1}

for p = 2,

H_{∗}(X; 𝔽_{p}) = 𝔽_{p}{x^{p−1}δx, β^{ϵ}Q^{i}(x) | i ≥ m + ϵ, β^{ϵ}Q^{i}(δx) | i ≥ m + r}

for p odd and t = 2m even, and

H_{∗}(X; 𝔽_{p}) = 𝔽_{p}{x(δx)^{p−1}, β^{ϵ}Q^{i}(x) | i ≥ m, β^{ϵ}Q^{i}(δx) | i ≥ m + r − 1 + ϵ}

for p odd and t = 2m − 1 odd. Throughout i is an integer and ϵ ∈ {0, 1}. The equivariant extension x′ induces an equivariant map

D_{p}(x′): X = D_{p}(S_{+}^{2r−1} ∧ S^{t}) → D_{p}(H ∧ R) .

The commutative S-algebra structures on H and R combine to form one on H ∧ R, and the associated H_{∞} structure includes, in particular, a 𝕋-equivariant structure map

ξ_{p}: D_{p}(H ∧ R) → H ∧ R

that extends the p-fold multiplication map on H ∧R. Taken together, these produce an equivariant map

H ∧ D_{p}(S_{+}^{2r−1} ∧ S^{t}) ⟶^{1∧D_{p}(x′)} H ∧ D_{p}(H ∧ R) ⟶^{1∧ξ_{p}} H ∧ H ∧ R ⟶^{μ∧1} H ∧ R ,

where μ is the multiplication on H. Applying homotopy we have a homomorphism

(5.2)  H_{∗}(X; 𝔽_{p}) = H_{∗}(D_{p}(S_{+}^{2r−1} ∧ S^{t}); 𝔽_{p}) → H_{∗}(R; 𝔽_{p})

which, by definition, takes the classes generating H_{∗}(X; 𝔽_{p}) to the classes with the same names in H_{∗}(R; 𝔽_{p}). Now X = D_{p}(S_{+}^{2r−1} ∧ S^{t}) is a 𝕋-equivariant retract of the free commutative S-algebra

P ≃ ⋁_{j≥0} D_{j}(S_{+}^{2r−1} ∧ S^{t})

on the space S_{+}^{2r−1} ∧S^{t}, so the homological homotopy fixed point spectral sequence for X is a direct summand of the one for P. Thus the formula from Proposition 4.1 for the d^{2r}-differentials in the spectral sequence for P is also applicable in the spectral sequence for X.

Now consider the homological homotopy fixed point spectral sequence for X = D_{p}(S_{+}^{2r−1} ∧ S^{t}), first for p = 2 and then for p odd. We shall show in each case that the 2r classes in E_{0,∗}^{2r} ⊂ H_{∗}(X; 𝔽_{p}), with names as listed in the statement of the theorem, are infinite cycles. By naturality of the homotopy fixed point spectral sequence with respect to the map H ∧ X → H ∧ R from (5.2), it follows that the 2r target classes listed in E_{0,∗}^{2r} ⊂ H_{∗}(R; 𝔽_{p}) are also infinite cycles. This will complete the proof of the theorem.

(a) Let p = 2. The homological homotopy fixed point spectral sequence for X has

E_{∗∗}^{2} = P (y) ⊗ 𝔽_{2}{xδx, Q^{i}(x) | i ≥ t, Q^{i}(δx) | i ≥ t + 2r − 1}

and nontrivial differentials d^{2r}(xδx) = y^{r} · (δx)^{2} and

d^{2r}(Q^{i}(x)) = y^{r} · Q^{i}(δx)

for all i ≥ t + 2r − 1, together with their y-multiples.

This leaves

E_{∗∗}^{2r+2} = P (y) ⊗ 𝔽_{2}{Q^{i}(x) | t ≤ i < t + 2r − 1, Q^{t+2r−1}(x) + xδx}

plus some y-torsion classes from E_{∗∗}^{2} in filtrations −2r < s ≤ 0. Hence there are no classes remaining in the entire quadrant with filtration s ≤ −2r and vertical degree ∗ > |xδx| = 2t + 2r − 1. All further differentials on the classes in E_{0,∗}^{2r+2} on the vertical axis land in this zero region, since already E_{0,∗}^{2} starts in degree 2t with the lowest class Q^{t}(x) = x^{2}. Thus all further differentials from the vertical axis are zero, and the spectral sequence collapses at E_{∗∗}^{2r+2} = E_{∗∗}^{∞}.

(b) Let p be odd and t = 2m even. The homological homotopy fixed point spectral sequence for X has

E_{∗∗}^{2} = P (y) ⊗ 𝔽_{p}{x^{p−1}δx, β^{ϵ}Q^{i}(x) | i ≥ m + ϵ, β^{ϵ}Q^{i}(δx) | i ≥ m + r}

and nontrivial differentials

d^{2r}(β^{ϵ}Q^{i}(x)) = y^{r} · β^{ϵ}Q^{i}(δx)

\xymatrix{y^{r} · Q^{t+2r}(δx) & ⋮ \\ y^{r} · (δx)^{2} & Q^{t+2r}(x) \ar@{|->}[ul]_{d^{2r}} \\  & Q^{t+2r−1}(x) \ar@{|->}[ul]^{d^{2r}} & xδx \ar@{|->}[ull]_{d^{2r}} \\  & Q^{t+2r−2}(x) \\  & ⋮ \\  & Q^{t}(x) = x^{2}}

Figure 1. The case p = 2.

for i ≥ m + r. This leaves

E_{∗∗}^{2r+2} = P (y) ⊗ 𝔽_{p}{x^{p−1}δx, Q^{i}(x) | m + ϵ ≤ i < m + r}

plus some y-torsion classes in filtrations −2r < s ≤ 0. Hence there are no classes left in the region where s ≤ −2r and the vertical degree is ∗ > |Q^{m+r−1}(x)|.

Now, x was also a class in the E^{2r−2}-term, with d^{2r−2}(x) = 0, so by induction over r we may assume (by naturality from the case of (r − 1)) that the classes β^{ϵ}Q^{i}(x) with m + ϵ ≤ i < m + (r − 1) are infinite cycles. This leaves the three classes x^{p−1}δx, βQ^{m+r−1}(x) and Q^{m+r−1}(x) in E_{0,∗}^{2r+2} that are not y-torsion, and could therefore imaginably support a differential after d^{2r}. But the first two classes βQ^{m+r−1}(x) and Q^{m+r−1}(x) are so close to the horizontal edge of the vanishing region that all differentials after d^{2r} must vanish on these classes.

The third class x^{p−1}δx has odd degree, so an even length differential on it must land in an even degree. The only even degree classes in filtrations s ≤ −2r are the y-multiples of Q^{i}(x) for m ≤ i < m+r, of which Q^{m}(x) = x^{p} is in lower degree than that of x^{p−1}δx. The remaining possible target classes Q^{i}(x) for m < i < m + r all have nontrivial Bockstein images βQ^{i}(x), but β(x^{p−1}δx) = 0 in H_{∗}(X; 𝔽_{p}). Therefore, by naturality of the differential with respect to the Bockstein operation, all of these targets for a differential on x^{p−1}δx are excluded. Thus also x^{p−1}δx is an infinite cycle.

(c) Let p be odd and t = 2m − 1 odd. The homological homotopy fixed point spectral sequence for X has

E_{∗∗}^{2} = P (y) ⊗ 𝔽_{p}{x(δx)^{p−1}, β^{ϵ}Q^{i}(x) | i ≥ m, β^{ϵ}Q^{i}(δx) | i ≥ m + r − 1 + ϵ}

\xymatrix{y^{r} · Q^{m+r}(δx) & ⋮ \\ y^{r} · βQ^{m+r}(δx) & Q^{m+r}(x) \ar@{|->}[ul]_{d^{2r}} \\  & βQ^{m+r}(x) \ar@{|->}[ul]_{d^{2r}} \\  & Q^{m+r−1}(x) \\  & βQ^{m+r−1}(x) \\  & ⋮ & x^{p−1}δx \\  & Q^{m}(x) = x^{p}}

Figure 2. The case p odd and t = 2m even.

and nontrivial differentials d^{2r}(x(δx)^{p−1}) = y^{r} · (δx)^{p} and

d^{2r}(β^{ϵ}Q^{i}(x)) = y^{r} · β^{ϵ}Q^{i}(δx)

for i ≥ m + r − 1 + ϵ. This leaves

E_{∗∗}^{2r+2} = P (y) ⊗ 𝔽_{p}{β^{ϵ}Q^{i}(x) | m ≤ i < m + r − 1 + ϵ, Q^{m+r−1}(x) − x(δx)^{p−1}}

plus y-torsion classes in filtrations −2r < s ≤ 0. Hence there are no classes left in the region where s ≤ −2r and the vertical degree is ∗ > |Q^{m+r−1}(x)|.

Again considering x as a class in E_{0,∗}^{2r−2} and using induction on r we may assume that the classes β^{ϵ}Q^{i}(x) for m ≤ i < m + r − 2 + ϵ and Q^{m+r−2}(x) − x(δ′x)^{p−1} are infinite cycles. Here δ′x is defined by d^{2r−2}(x) = y^{r−1} · δ′x. The fact that d^{2r−2}(x) = 0 gives δ′x = 0, so in fact all the classes β^{ϵ}Q^{i}(x) for m ≤ i < m + r − 1 in E_{∗∗}^{2r+2} are infinite cycles.

This leaves only the two classes βQ^{m+r−1}(x) and Q^{m+r−1}(x) − x(δx)^{p−1}, but these are so close to the horizontal border of the vanishing region that all differentials after d^{2r} must be zero on them. □

6. Examples

Our Theorem 5.1 has applications to the homological homotopy fixed point spectral sequence for the commutative S-algebra R = T HH(B) given by the topological

\xymatrix{y^{r} · βQ^{m+r}(δx) & ⋮ \\ y^{r} · (δx)^{p} & βQ^{m+r}(x) \ar@{|->}[ul]_{d^{2r}} \\  & Q^{m+r−1}(x) \ar@{|->}[ul]^{d^{2r}} & x(δx)^{p−1} \ar@{|->}[ull]_{d^{2r}} \\  & βQ^{m+r−1}(x) \\  & ⋮ \\  & βQ^{m}(x)}

Figure 3. The case p odd and t = 2m − 1 odd.

Hochschild homology of a commutative S-algebra B. The 𝕋-homotopy fixed point spectrum T HH(B)^{h𝕋} is closely related to the topological model T F (B) for the negative cyclic homology of B, which in turn is very close to the topological cyclic homology T C(B) [BHM93] and algebraic K-theory K(B) of B [Du97]. These spectral sequences therefore have significant interest.

First consider the connective Johnson–Wilson spectra B = BP ⟨m − 1⟩, for some prime p and integer 0 ≤ m < ∞. So

π_{∗}BP ⟨m − 1⟩ = BP_{∗}/(v_{n} | n ≥ m) ,

where BP_{∗} = ℤ_{(p)}[v_{n} | n ≥ 1] and v_{0} = p, and

H_{∗}(BP ⟨m − 1⟩; 𝔽_{p}) = \{ \begin{matrix}P (ξ̄_{1}^{2}, …, ξ̄_{m}^{2}, ξ̄_{m+1}, …) & for p = 2, \\ P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ m) & for p odd. \end{matrix}

The latter is a sub-algebra of the dual Steenrod algebra A_{∗} = H_{∗}(H𝔽_{p}; 𝔽_{p}) [Mi58].

Suppose that p and m are such that BP ⟨m − 1⟩ admits the structure of a commutative S-algebra. This is so at least for m ∈ {0, 1, 2}, when BP ⟨−1⟩ = H𝔽_{p}, BP ⟨0⟩ = Hℤ_{(p)} and BP ⟨1⟩ = ℓ, respectively, where ℓ is the Adams summand of p-local connective topological K-theory ku_{(p)}. When p = 2, ℓ = ku_{(2)}.

Then the Bökstedt spectral sequence

E_{∗∗}^{2} = HH_{∗}(H_{∗}(B; 𝔽_{p})) ⟹ H_{∗}(T HH(B); 𝔽_{p})

has E^{2}-term

E_{∗∗}^{2} = \{ \begin{matrix}H_{∗}(BP ⟨m − 1⟩; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, …, σξ̄_{m}^{2}, σξ̄_{m+1}, …) & for p = 2, \\ H_{∗}(BP ⟨m − 1⟩; 𝔽_{p}) ⊗ E(σξ̄_{k} | k ≥ 1) ⊗ Γ(στ̄_{k} | k ≥ m) & for p odd. \end{matrix}

For x ∈ H_{∗}(B; 𝔽_{p}), σx ∈ HH_{1}(H_{∗}(B; 𝔽_{p})) is represented by the Hochschild 1cycle 1 ⊗ x. The operator σ is a differential (σ^{2} = 0) and a graded derivation (σ(xy) = xσ(y) + (−1)^{|y|}σ(x)y). Here Γ(−) denotes the divided power algebra.

For p odd, Bökstedt found differentials

d^{p−1}(γ_{j}(στ̄_{k})) = σξ̄_{k+1} · γ_{j−p}(στ̄_{k})

for j ≥ p, and in all cases the spectral sequence collapses at the E^{p}-term. So

E_{∗∗}^{∞} = \{ \begin{matrix}H_{∗}(BP ⟨m − 1⟩; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, …, σξ̄_{m}^{2}, σξ̄_{m+1}, …) & for p = 2, \\ H_{∗}(BP ⟨m − 1⟩; 𝔽_{p}) ⊗ E(σξ̄_{1}, …, σξ̄_{m}) ⊗ P_{p}(στ̄_{k} | k ≥ m) & for p odd. \end{matrix}

Here P_{p}(−) denotes the truncated polynomial algebra of height p.

If BP ⟨m − 1⟩, and thus T HH(BP ⟨m − 1⟩), is a commutative S-algebra, then (σξ̄_{k})^{2} = σξ̄_{k+1} for p = 2 and (στ̄_{k})^{p} = στ̄_{k+1} for p odd, so

H_{∗}(T HH(BP ⟨m − 1⟩); 𝔽_{p})

= \{ \begin{matrix}H_{∗}(BP ⟨m − 1⟩; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, …, σξ̄_{m}^{2}) ⊗ P (σξ̄_{m+1}) & for p = 2, \\ H_{∗}(BP ⟨m − 1⟩; 𝔽_{p}) ⊗ E(σξ̄_{1}, …, σξ̄_{m}) ⊗ P (στ̄_{m}) & for p odd. \end{matrix}

For more references and details on the calculation up to this point, see [AnR, §5].

We now consider the homological homotopy fixed point spectral sequence for R = T HH(B). It starts with

E_{∗∗}^{2} = P (y) ⊗ H_{∗}(T HH(B); 𝔽_{p})

and by Lemma 3.1 it has first differentials

d^{2}(x) = y · σx

for all x ∈ H_{∗}(T HH(B); 𝔽_{p}). Here σx ∈ H_{t+1}(T HH(B); 𝔽_{p}) is the image of x ⊗ s_{1} ∈ H_{t}(T HH(B); 𝔽_{p}) ⊗ H_{1}(𝕋; 𝔽_{p}) under the circle action map

α: T HH(B) ∧ 𝕋_{+} → T HH(B) ,

where s_{1} ∈ H_{1}(𝕋; 𝔽_{p}) is the canonical generator. By Lemma 4.3 we have similar differentials d^{2}(y^{n} · x) = y^{n+1} · σx for all n ≥ 0.

Hence we can find the columns of E_{∗∗}^{4} in the homological homotopy fixed point spectral sequence by passing to the homology of E_{0,∗}^{2} = H_{∗}(T HH(B); 𝔽_{p}) with respect to the operator σ, at least to the left of the vertical axis.

Proposition 6.1. The homological homotopy fixed point spectral sequence for R = T HH(B) with B = BP ⟨m − 1⟩, for p and m such that B is a commutative Salgebra, collapses after the d^{2}-differentials, with the following E^{∞}-term:

(a) For p = 2,

E_{∗∗}^{∞} = P (y) ⊗ P (ξ̄_{1}^{4}, …, ξ̄_{m}^{4}, ξ̄_{m+1}^{2}, ξ′_{m+2}, …) ⊗ E(ξ̄_{1}^{2}σξ̄_{1}^{2}, …, ξ̄_{m}^{2}σξ̄_{m}^{2})

plus some classes in filtration s = 0, where ξ′_{k+1} = ξ̄_{k+1} + ξ̄_{k}σξ̄_{k} for k ≥ m + 1. (b) For p odd,

E_{∗∗}^{∞} = P (y) ⊗ P (ξ̄_{k}^{p} | 1 ≤ k ≤ m) ⊗ P (ξ̄_{k+1} | k ≥ m)

⊗ E(τ′_{k+1} | k ≥ m) ⊗ E(ξ̄_{k}^{p−1}σξ̄_{k} | 1 ≤ k ≤ m)

plus some classes in filtration s = 0, where τ′_{k+1} = τ̄_{k+1} − τ̄_{k}(στ̄_{k})^{p−1} for k ≥ m.

Proof. (a) For B = BP ⟨m − 1⟩ and p = 2 we have

E_{0,∗}^{2} = H_{∗}(T HH(BP ⟨m − 1⟩); 𝔽_{2})

= P (ξ̄_{1}^{2}, …, ξ̄_{m}^{2}, ξ̄_{m+1}, …) ⊗ E(σξ̄_{1}^{2}, …, σξ̄_{m}^{2}) ⊗ P (σξ̄_{m+1}) .

Here σ: ξ̄_{k}^{2} ↦ σξ̄_{k}^{2} for 1 ≤ k ≤ m and σ: ξ̄_{k+1} ↦ σξ̄_{k+1} for k ≥ m. We have σξ̄_{k+1} = (σξ̄_{k})^{2} for k ≥ m + 1.

So the squares (ξ̄_{k}^{2})^{2} = ξ̄_{k}^{4} and ξ̄_{m+1}^{2}, as well as the companion classes defined by

ξ′_{k+1} = ξ̄_{k+1} + ξ̄_{k}σξ̄_{k}

for k ≥ m + 1, are d^{2}-cycles, while E(ξ̄_{k}^{2}, σξ̄_{k}^{2}) has homology E(ξ̄_{k}^{2}σξ̄_{k}^{2}) for each k, and E(ξ̄_{m+1}) ⊗ P (σξ̄_{m+1}) has homology 𝔽_{2}.

Hence the homological spectral sequence has

E_{∗∗}^{4} = P (y) ⊗ P (ξ̄_{1}^{4}, …, ξ̄_{m}^{4}, ξ̄_{m+1}^{2}, ξ′_{m+2}, …) ⊗ E(ξ̄_{1}^{2}σξ̄_{1}^{2}, …, ξ̄_{m}^{2}σξ̄_{m}^{2})

plus the image of σ in filtration s = 0.

By our Theorem 5.1(a) applied to the classes x = ξ̄_{k}^{2} for 1 ≤ k ≤ m, in even degree t = |x|, the classes x^{2} = ξ̄_{k}^{4} and Q^{t+1}(x) + xσx = ξ̄_{k}^{2}σξ̄_{k}^{2} are infinite cycles, for Q^{t+1}(ξ̄_{k}^{2}) = 0 by the Cartan formula.

Similarly, by Theorem 5.1(a) applied to the classes x = ξ̄_{k} for k ≥ m + 1, in odd degree t = |x|, the classes x^{2} = ξ̄_{k}^{2} and Q^{t+1}(x) + xσx = ξ̄_{k+1} + ξ̄_{k}σξ̄_{k} = ξ′_{k+1} are infinite cycles. For Q^{t+1}(ξ̄_{k}) = ξ̄_{k+1} by [BMMS86, III.2.2 and I.3.6].

The extra classes in filtration s = 0 are y-torsion, hence infinite cycles. Therefore the E^{4}-term above is generated as an algebra by infinite cycles, so the homological spectral sequence collapses at this stage.

(b) For B = BP ⟨m − 1⟩ and p odd we have

E_{0,∗}^{2} = H_{∗}(T HH(BP ⟨m − 1⟩); 𝔽_{p})

= P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ m) ⊗ E(σξ̄_{k} | 1 ≤ k ≤ m) ⊗ P (στ̄_{m}) .

Here σ: ξ̄_{k} ↦ σξ̄_{k} for 1 ≤ k ≤ m, σ: ξ̄_{k+1} ↦ 0 for k ≥ m and σ: τ̄_{k} ↦ στ̄_{k} for k ≥ m. We have στ̄_{k+1} = (στ̄_{k})^{p} for k ≥ m.

So the p-th powers ξ̄_{k}^{p} for 1 ≤ k ≤ m, the classes ξ̄_{k+1} for k ≥ m, and the companion classes defined by

τ′_{k+1} = τ̄_{k+1} − τ̄_{k}(στ̄_{k})^{p−1}

for k ≥ m, are d^{2}-cycles, while P_{p}(ξ̄_{k}) ⊗ E(σξ̄_{k}) has homology E(ξ̄_{k}^{p−1}σξ̄_{k}) for each 1 ≤ k ≤ m, and E(τ̄_{m}) ⊗ P (στ̄_{m}) has homology 𝔽_{p}.

Hence the homological spectral sequence has

E_{∗∗}^{4} = P (y) ⊗ P (ξ̄_{k}^{p} | 1 ≤ k ≤ m) ⊗ P (ξ̄_{k+1} | k ≥ m)

⊗ E(τ′_{k+1} | k ≥ m) ⊗ E(ξ̄_{k}^{p−1}σξ̄_{k} | 1 ≤ k ≤ m)

plus some classes in filtration s = 0.

Applying our Theorem 5.1(b) to the classes x = ξ̄_{k} for 1 ≤ k ≤ m, in even degree t = |x|, the classes x^{p} = ξ̄_{k}^{p} and x^{p−1}σx = ξ̄_{k}^{p−1}σξ̄_{k} are infinite cycles.

Similarly, applying Theorem 5.1(c) to the classes x = τ̄_{k} for k ≥ m, in odd degree t = |x| = 2p^{k} − 1, the classes βQ^{p^{k}}(x) = ξ̄_{k+1} and Q^{p^{k}}(x) − x(σx)^{p−1} = τ̄_{k+1} − τ̄_{k}(στ̄_{k})^{p−1} = τ′_{k+1} are infinite cycles, for Q^{p^{k}}(τ̄_{k}) = τ̄_{k+1} and βτ̄_{k+1} = ξ̄_{k+1} by [BMMS86, III.2.3 and I.3.6].

Hence the E^{4}-term above is generated as an algebra by infinite cycles, and the homological spectral sequence collapses after the d^{2}-differentials. □

For convenience in the comparison with ko, we make the case B = ku at p = 2 explicit:

Corollary 6.2. The homological homotopy fixed point spectral sequence for R = T HH(ku) at p = 2 collapses after the d^{2}-differentials, with

E_{∗∗}^{∞} = P (y) ⊗ P (ξ̄_{1}^{4}, ξ̄_{2}^{4}, ξ̄_{3}^{2}, ξ′_{4}, …) ⊗ E(ξ̄_{1}^{2}σξ̄_{1}^{2}, ξ̄_{2}^{2}σξ̄_{2}^{2})

plus some classes in filtration s = 0, where ξ′_{k+1} = ξ̄_{k+1} + ξ̄_{k}σξ̄_{k} for k ≥ 3.

Proposition 6.3. The homological homotopy fixed point spectral sequence for R = T HH(B) collapses after the d^{2}-differentials, in both of the cases:

(a) B = ko and p = 2, when

E_{∗∗}^{∞} = P (y) ⊗ P (ξ̄_{1}^{8}, ξ̄_{2}^{4}, ξ̄_{3}^{2}, ξ′_{4}, …) ⊗ E(ξ̄_{1}^{4}σξ̄_{1}^{4}, ξ̄_{2}^{2}σξ̄_{2}^{2})

plus classes on the vertical axis, and (b) B = tmf and p = 2, when

E_{∗∗}^{∞} = P (y) ⊗ P (ξ̄_{1}^{16}, ξ̄_{2}^{8}, ξ̄_{3}^{4}, ξ̄_{4}^{2}, ξ′_{5}, …) ⊗ E(ξ̄_{1}^{8}σξ̄_{1}^{8}, ξ̄_{2}^{4}σξ̄_{2}^{4}, ξ̄_{3}^{2}σξ̄_{3}^{2})

plus classes on the vertical axis.

Proof. (a) For B = ko with H_{∗}(B; 𝔽_{2}) = (A//A_{1})_{∗} = P (ξ̄_{1}^{4}, ξ̄_{2}^{2}, ξ̄_{3}, …) we have

H_{∗}(T HH(ko); 𝔽_{2}) = P (ξ̄_{1}^{4}, ξ̄_{2}^{2}, ξ̄_{3}, …) ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3}) .

See [AnR, 6.2(a)].

As in the proof of Proposition 6.1, the squares ξ̄_{1}^{8}, ξ̄_{2}^{4} and ξ̄_{3}^{2}, as well as the classes ξ′_{k+1} = ξ̄_{k+1} + ξ̄_{k}σξ̄_{k} for k ≥ 3 are d^{2}-cycles, while E(ξ̄_{1}^{4}, σξ̄_{1}^{4}) and E(ξ̄_{2}^{2}, σξ̄_{2}^{2}) have homology E(ξ̄_{1}^{4}σξ̄_{1}^{4}) and E(ξ̄_{2}^{2}σξ̄_{2}^{2}), respectively. The homology of E(ξ̄_{3}) ⊗ P (σξ̄_{3}) is 𝔽_{2}. So

E_{∗∗}^{4} = P (y) ⊗ P (ξ̄_{1}^{8}, ξ̄_{2}^{4}, ξ̄_{3}^{2}, ξ′_{4}, …) ⊗ E(ξ̄_{1}^{4}σξ̄_{1}^{4}, ξ̄_{2}^{2}σξ̄_{2}^{2})

plus some classes in filtration s = 0.

By Theorem 5.1(a), all of these algebra generators are in fact infinite cycles, so the homological spectral sequence collapses, as claimed.

(b) For B = tmf with H_{∗}(B; 𝔽_{2}) = (A//A_{2})_{∗} = P (ξ̄_{1}^{8}, ξ̄_{2}^{4}, ξ̄_{3}^{2}, ξ̄_{4}, …) we have

H_{∗}(T HH(tmf); 𝔽_{2}) = P (ξ̄_{1}^{8}, ξ̄_{2}^{4}, ξ̄_{3}^{2}, ξ̄_{4}, …) ⊗ E(σξ̄_{1}^{8}, σξ̄_{2}^{4}, σξ̄_{3}^{2}) ⊗ P (σξ̄_{4}) .

See [AnR, 6.2(b)]. This gives the E^{2}-term of the homological spectral sequence, and as before its homology with respect to the σ-operator is

E_{∗∗}^{4} = P (y) ⊗ P (ξ̄_{1}^{16}, ξ̄_{2}^{8}, ξ̄_{3}^{4}, ξ̄_{4}^{2}, ξ′_{5}, …) ⊗ E(ξ̄_{1}^{8}σξ̄_{1}^{8}, ξ̄_{2}^{4}σξ̄_{2}^{4}, ξ̄_{3}^{2}σξ̄_{3}^{2})

plus some classes in filtration s = 0.

By Theorem 5.1(a), all of these algebra generators are in fact infinite cycles, so the homological spectral sequence collapses, as claimed. □

Theorem 6.4. The homological homotopy fixed point spectral sequence for R = T HH(B) collapses after the d^{2}-differentials, in both of the cases:

(a) B = MU, with

E_{∗∗}^{∞} = P (y) ⊗ P (b_{k}^{p} | k ≥ 1) ⊗ E(b_{k}^{p−1}σb_{k} | k ≥ 1)

plus classes in filtration zero, and (b) B = BP, with

E_{∗∗}^{∞} = P (y) ⊗ P (ξ_{k}^{p} | k ≥ 1) ⊗ E(ξ_{k}^{p−1}σξ_{k} | k ≥ 1)

plus classes in filtration zero. (When p = 2, substitute ξ_{k}^{2} for ξ_{k}.)

Note that we do not need to assume that BP is a commutative S-algebra for the result in part (b).

Proof. The integral homology algebra of MU is H_{∗}(MU; ℤ) = ℤ[b_{k} | k ≥ 1], where b_{k} in degree 2k is the stabilized image of the generator β_{k+1} ∈ H_{2k+2}(BU(1); ℤ), under the zero-section identification BU(1) ≃ MU(1). So

H_{∗}(MU; 𝔽_{p}) = P (b_{k} | k ≥ 1)

is concentrated in even degrees, and the E^{2}-term of the Bökstedt spectral sequence is

E_{∗∗}^{2} = HH_{∗}(H_{∗}(MU; 𝔽_{p})) = H_{∗}(MU; 𝔽_{p}) ⊗ E(σb_{k} | k ≥ 1) .

All the algebra generators are in filtrations s ≤ 1, so the spectral sequence collapses at this stage. There are no algebra extensions, since for p = 2, (σb_{k})^{2} = Q^{2k+1}(σb_{k}) = σQ^{2k+1}(b_{k}) = 0, where Q^{2k+1}(b_{k}) = 0 because it has odd degree. For p odd, (σb_{k})^{2} = 0 by graded commutativity, because σb_{k} has odd degree. Thus

H_{∗}(T HH(MU); 𝔽_{p}) = H_{∗}(MU; 𝔽_{p}) ⊗ E(σb_{k} | k ≥ 1) .

This much can also be read off from [MS93, 4.3], or from Cohen and Schlichtkrull’s formula T HH(MU) ≃ MU ∧ SU_{+} [CS].

The homological homotopy fixed point spectral sequence has E^{2}-term

E_{∗∗}^{2} = P (y) ⊗ P (b_{k} | k ≥ 1) ⊗ E(σb_{k} | k ≥ 1) .

Its homology with respect to the d^{2}-differential, satisfying d^{2}(b_{k}) = y · σb_{k}, is

E_{∗∗}^{4} = P (y) ⊗ P (b_{k}^{p} | k ≥ 1) ⊗ E(b_{k}^{p−1}σb_{k} | k ≥ 1)

plus the usual y-torsion on the vertical axis. By Theorem 5.1(a) and (b), the algebra generators of this E^{4}-term are all infinite cycles. Hence the spectral sequence collapses at this stage.

(b) The Brown–Peterson spectrum BP was originally constructed to have mod p homology

H_{∗}(BP; 𝔽_{p}) = \{ \begin{matrix}P (ξ_{k}^{2} | k ≥ 1) & for p = 2, \\ P (ξ_{k} | k ≥ 1) & for p odd. \end{matrix}

This equals the sub-algebra (A//E)_{∗} of A_{∗} that is dual to the quotient algebra A//E = A/AβA of A. Hereafter we focus on the odd-primary case; the reader should substitute ξ_{k}^{2} for ξ_{k} when p = 2.

The spectrum BP is known to be an (associative) S-algebra, and to receive an S-algebra map from MU [BJ02, 3.5]. This map induces a split surjective algebra homomorphism H_{∗}(MU; 𝔽_{p}) → H_{∗}(BP; 𝔽_{p}) in homology, which maps b_{p^{k}−1} to ξ_{k} for k ≥ 1 and takes the remaining algebra generators b_{i} to 0 for i ≠ p^{k} − 1. For the homology of BP injects into H_{∗}(Hℤ_{(p)}; 𝔽_{p}) and at the level of second spaces the composite map of spectra MU → BP → Hℤ_{(p)} is a p-local equivalence MU(1) → K(ℤ_{(p)}, 2). The generator β_{i+1} ∈ H̃_{2i+2}(MU(1); 𝔽_{p}) maps to b_{i} ∈ H_{2i}(MU; 𝔽_{p}), while the corresponding generator β_{i+1} ∈ H̃_{2i+2}(K(ℤ_{(p)}, 2); 𝔽_{p}) maps to ξ_{k} ∈ H_{2i}(Hℤ_{(p)}; 𝔽_{p}) when i = p^{k} − 1 and to 0 otherwise [Mi58, §5]. This proves the claim.

The Bökstedt spectral sequence for BP has E^{2}-term

E_{∗∗}^{2} = HH_{∗}(H_{∗}(BP; 𝔽_{p})) = H_{∗}(BP; 𝔽_{p}) ⊗ E(σξ_{k} | k ≥ 1) .

Note that the map MU → BP induces a surjection of Bökstedt spectral sequence E^{2}-terms. Thus the fact that the Bökstedt spectral sequence for MU collapses at E^{2} with no algebra extensions implies the corresponding statement for BP, also without the assumption that BP is a commutative S-algebra. We can conclude that

H_{∗}(T HH(BP); 𝔽_{p}) = H_{∗}(BP; 𝔽_{p}) ⊗ E(σξ_{k} | k ≥ 1) .

The homological homotopy fixed point spectral sequence has E^{2}-term

E_{∗∗}^{2} = P (y) ⊗ P (ξ_{k} | k ≥ 1) ⊗ E(σξ_{k} | k ≥ 1) .

Again the map MU → BP induces a surjection of E^{2}-terms, so the d^{2}-differentials satisfy d^{2}(ξ_{k}) = y · σξ_{k} and d^{2}(y) = 0, and are derivations. This leaves

E_{∗∗}^{4} = P (y) ⊗ P (ξ_{k}^{p} | k ≥ 1) ⊗ E(ξ_{k}^{p−1}σξ_{k} | k ≥ 1)

plus some y-torsion on the vertical axis, and the map from the E^{4}-term of the spectral sequence for MU is still surjective. Thus the spectral sequence for BP also collapses at this stage. □

7. Generalizations and comments

In this section we note some generalizations of our results, and also comment on the relation to related patterns of differentials in other spectral sequences. The generalizations are of two sorts. First, we can replace the homotopy fixed points construction by the Tate construction or the homotopy orbits. Second, we can change the group of equivariance. We consider these in order.

First, there are spectral sequences similar to the one considered here for the Tate construction X^{t𝕋} = [Ẽ𝕋 ∧ F (E𝕋_{+}, X)]^{𝕋} (denoted t_{𝕋}(X)^{𝕋} in [GM95] and ℍ̂(𝕋, X) in [AuR02]) and the homotopy orbit spectrum X_{h𝕋} = E𝕋_{+} ∧_{𝕋} X.

Proposition 7.1. There is a natural spectral sequence

Ê_{∗∗}^{2} = Ĥ^{−∗}(𝕋; H_{∗}(X; 𝔽_{p})) = P (y, y^{−1}) ⊗ H_{∗}(X; 𝔽_{p})

with y in bidegree (−2, 0), which is conditionally convergent to the continuous homology H_{∗}^{c}(X^{t𝕋}; 𝔽_{p}). We call this the homological Tate spectral sequence. If H_{∗}(X; 𝔽_{p}) is finite in each degree, or the spectral sequence collapses at a finite stage, then the spectral sequence is strongly convergent.

Proposition 7.2. There is a natural spectral sequence

E_{∗∗}^{2} = H_{∗}(𝕋; H_{∗}(X; 𝔽_{p})) = P (y^{−1}) ⊗ H_{∗}(X; 𝔽_{p})

with y^{−1} in bidegree (2, 0), which is strongly convergent to H_{∗}(X_{h𝕋}; 𝔽_{p}). We call this the homological homotopy orbit spectral sequence. (Note that for X_{h𝕋} the continuous homology is the same as the ordinary homology.)

Further, the middle and right hand maps of the (homotopy) norm cofiber sequence

ΣX_{h𝕋} ⟶^{N} X^{h𝕋} → X^{t𝕋} → Σ^{2}X_{h𝕋}

induce the homomorphisms of E^{2}-terms given by tensoring H_{∗}(X; 𝔽_{p}) with the short exact sequence of P (y)-modules

0 → P (y) → P (y, y^{−1}) → Σ^{2}P (y^{−1}) → 0.

Thus the homological Tate spectral sequence is an upper half plane spectral sequence whose E^{2}-term is obtained by continuing the y-periodicity in the homological homotopy fixed point spectral sequence into the first quadrant, and the homological homotopy orbit spectral sequence (shifted 2 degrees to the right from Proposition 7.2) has the quotient of these as its E^{2}-term.

Proposition 4.1 and Theorem 5.1 apply equally well to all three spectral sequences. For details, see the thesis of Lunøe–Nielsen [L-N].

Second, we could also consider these three spectral sequences for the action of a finite cyclic subgroup C of 𝕋. For example, there is the homological Tate spectral sequence

Ê_{∗∗}^{2} = Ĥ^{−∗}(C; H_{∗}(X; 𝔽_{p}))

converging conditionally to H_{∗}^{c}(X^{tC}; 𝔽_{p}). The analogue of Lemma 4.3 still holds, so that there are isomorphisms

Ê_{∗∗}^{r} ≅ Ĥ^{−∗}(C; 𝔽_{p}) ⊗ Ê_{0,∗}^{r}

for all r ≥ 2 (and now y is invertible, so there is no y-torsion), and all differentials are determined by those originating on the vertical axis Ê_{0,∗}^{r}. In turn, the latter differentials are determined by those in the 𝕋-equivariant case, by naturality with respect to the restriction map X^{t𝕋} → X^{tC}. Therefore the collapsing results in Theorem 5.1 also hold in these cases. See [L-N] for more details.

These latter spectral sequences, for finite subgroups C ⊂ 𝕋, are essential in the analysis of the topological model T F (B) for the negative cyclic homology of B, and the topological cyclic homology T C(B).

Though the differentials here allow us to determine E_{∗∗}^{∞} in the cases of interest (see Section 6), there are still A_{∗}-comodule extensions hidden by the filtration. These are of course of critical importance for the analysis of the Adams spectral sequence (1.3). A more elaborate study of the geometry of the universal examples used in Section 5 allows these to be recovered. This too can be found in [L-N].

Finally, it is interesting to compare the formulas for differentials here to analogous results in other spectral sequences. The first to be considered was the Adams spectral sequence, where the results are due to Kahn [Ka70], Milgram [Mi72], Mäkinen [Mä73], and the first author [BMMS86, Ch. VI]. For simplicity, let us assume p = 2 in this discussion, as there are several cases to be considered at odd primes ([BMMS86, VI.1.1]). Suppose that x is in the E_{r}-term of the Adams spectral sequence

E_{2}^{∗∗} = Ext_{A}(H^{∗}(R; 𝔽_{2}), 𝔽_{2}) ⟹ π_{∗}(R)_{2}^{∧} ,

where R is a commutative S-algebra. The commutative S-algebra structure of R induces Steenrod operations in the E_{2}-term of the Adams spectral sequence, which are the analog in this situation of the Dyer–Lashof operations in H_{∗}(R; 𝔽_{2}). (In fact, under the Hurewicz homomorphism, they map to the Dyer–Lashof operations.) Then, in most cases we have

(7.3)  d_{∗}(Sq^{j}x) = Sq^{j}d_{r}(x) ˙+ aSq^{j−v}x ,

where A ˙+ B denotes whichever of A or B is in the lower filtration, or their sum, if they are in the same filtration. The subscript in d_{∗} is then the difference in filtrations between the right and left hand sides. In this formula, a is an infinite cycle in the Adams spectral sequence for the homotopy groups of spheres, and a and v are determined by j and the degree of x. When the first half of the right hand side dominates we have

d_{2r−1}(Sq^{j}x) = Sq^{j}d_{r}(x) ,

and this formula resembles the formula

d^{2r}(β^{ϵ}Q^{i}(x)) = β^{ϵ}Q^{i}(d^{2r}(x))

of Proposition 4.1, in that both essentially say that the relevant differential commutes with the Dyer–Lashof operations. The fact that the length of the differential increases from r to (2r − 1) when we apply the squaring operation in the Adams spectral sequence reflects the difference between the homotopy fixed point filtration and the Adams filtration, and the way in which they interact with the extended powers. A more extreme difference occurs when the second term aSq^{j−v}x is involved. In the homological homotopy fixed point spectral sequence this term disappears, essentially because the element a ∈ π_{∗}S is mapped to 0 by the Hurewicz homomorphism. Homotopical homotopy fixed point spectral sequences, as in [AuR02], will have differential formulas with two parts, as in the Adams spectral sequence. Such two part formulas for differentials reflect universally hidden extensions in the following sense.

The differential (7.3) arises from decomposing the boundary of the cell on which Sq^{j}x is defined into two pieces. One of the pieces carries Sq^{j}d_{r}(x) and the other carries aSq^{j−v}x. The half that lies in the lower filtration is killed by the differential (7.3), and therefore appears to be 0 in the associated graded E_{∞}. However, the geometry of the situation shows that it is actually equal to the half of the formula that lies in the higher filtration, modulo still higher filtrations. Thus we have a universally hidden extension, that is, an expression which is 0 in the associated graded, by virtue of being equal to an expression which lies in a higher filtration. We should expect this sort of phenomenon to occur in homotopical homotopy fixed point spectral sequences.

Finally, Theorem 5.1 seems to be particular to the homological homotopy fixed point spectral sequence. Certainly the Adams spectral sequence seems to have no analog of this extreme cutoff, in which certain terms die at E^{r} and the remaining terms live to E^{∞}.

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Department of Mathematics, Wayne State University, USA

   E-mail address: rrb@math.wayne.edu

Department of Mathematics, University of Oslo, Norway

   E-mail address: rognes@math.uio.no